The order of an ODE is the highest derivative in the equation. An n-th order ODE can be reduced to a system of first order ODEs. Hence the solution of ODEs of any order can be seen as the solution of a system of first order ODEs:

Boundary Conditions

The nature of the boundary condition is critical
in the determination of the numerical method that
is to employed in its solution. For example
the initial value problem the conditions at
t=a are given. At two-point value problems
conditions must be applied at both x=a and
x=b.

Euler's Method

By using the simple approximation (x(c+h)-x(c))/h
for dx(c)/dt for a fixed step length h and starting
from t=a and the initial value x(a), the solution
at each step ( x(a+h), x(a+2h) etc) can be found
through the substitution of this expression in the
original ODE. Although this method does
work it is found to be very inefficient in comparison with
alternative methods and it is seldom used in practice.
Worksheet and Examples in Excel

Runge-Kutta Methods

In Runge-Kutta methods the integration interval
[c,c+h] is divided into sub-intervals, the derivative
is found at each of the subintervals and a
weighted average is taken to obtain an accurate solution.
Worksheet and Examples in Excel

Predictor Corrector Methods

Predictor-corrector methods are methods in which two
formulas are used one to "predict" the solution
at each step, the corrector also uses the predicted
value in its formula to "correct" the solution.

Two-point BVP

Such problems can be solved by the shooting method
in which values for all conditions that are
not given at x=a are estimated. A method described above is
then employed to find the solution. The results at
x=b are then compared with the actual boundary condition.
With this knowledge, the conditions at x=a are
changed and the solution run again. This continues
until a satisfactory result is obtained.

Alternatively, relaxation methods can be used. The ODE is replaced by a difference equation defined throughout the range [a,b]. Starting with a trial solution, an iterative process called relaxation is then employed to bring the numerical solution closer and closer to satisfying the ODE and the boundary conditions. In this method Books on the methods described above are listed in www.science-books.net . Click on the topic of interest below.

[mathematics.me.uk] [computing.me.uk] [engineering.me.uk] [physics.me.uk] [statistics.me.uk]